## Reasoning about Finite-State Switched Systems

### BibTeX

@MISC{Fisman_reasoningabout,

author = {Dana Fisman and Orna Kupferman},

title = {Reasoning about Finite-State Switched Systems},

year = {}

}

### OpenURL

### Abstract

Abstract. A switched system is composed of components. The components do not interact with one another. Rather, they all interact with the same environment, which switches one of them on at each moment in time. In standard concurrency, a component restricts the environment of the other components, thus the concurrent system has fewer behaviors than its components. On the other hand, in a switched system, a component suggests an alternative to the other components, thus the switched system has richer behaviors than its components. We study finite-state switched systems, where each of the underlying components is a finite-state transducer. While the main challenge, namely compositionality, is similar in standard concurrent systems and in switched systems, the problems and solutions are different. In the verification front, we suggest and study an assumeguarantee paradigm for switched systems, and study formalisms in which satisfaction of a specification in all components imply its satisfaction in the switched system. In the synthesis front, we show that while compositional synthesis and design are undecidable, the problem of synthesizing a switching rule with which a given switched system satisfies an LTL specification is decidable. 1

### Citations

357 |
On the synthesis of a reactive module
- Pnueli, Rosner
- 1989
(Show Context)
Citation Context ...thesizing a switching rule according to which the switching system satisfies a specification. We show that the problem has the same flavor as the standard LTL control problem, and is 2EXPTIMEcomplete =-=[17]-=-. The solution to the problem, however, is different, as the synthesized switched rule does not disable transitions, as is the case in usual control. Rather, it chooses the component that is switched ... |

264 | Reasoning about infinite computations
- Vardi, Wolper
- 1994
(Show Context)
Citation Context ...es ψ. We reduced the switching-rule synthesis problem to the nonemptiness problem for . The number of states of the DPW Aψ is doubly-exponential in |ψ|, and its A T1,T2 ∀ψ index is exponential in |ψ| =-=[20, 23]-=-. Therefore, the number of states of the DPT A T1,T2 ∀ψ is linear in |T1| and |T2| and doubly-exponential in |ψ|, and its index is exponential in |ψ|. Since the nonemptiness problem for DPT can be sol... |

239 |
Efficient model checking in fragments of the propositional mu-calculus
- Emerson, Lei
- 1986
(Show Context)
Citation Context ...1| and |T2| and doubly-exponential in |ψ|, and its index is exponential in |ψ|. Since the nonemptiness problem for DPT can be solved in time polynomial in the state space and exponential in the index =-=[5]-=-, the upper bound follows. Note that the doubly-exponential complexity is only in terms of |ψ|, and the algorithm is polynomial in |T1| and |T2|. For the lower bound, note that the synthesis problem f... |

143 | Qualitative simulation of genetic regulatory networks using piecewise-linear models
- JONG, GOUZÉ, et al.
(Show Context)
Citation Context ...ems (c.f., internet communication protocols [7, 22]), mechanical systems (engines with gear transmission [8]), electrical circuits (power converters [6]), biological systems (gene regulating networks =-=[3]-=-), and embedded systems combining the above [2]. There has been extensive research in the control engineering community on analysis of continuous switched systems whose evolution is described by means... |

142 | Channel models for ultrawideband personal area networks
- Molisch
- 2003
(Show Context)
Citation Context ...s systems enables reasoning about other aspects of systems. For example, consider a cell phone that may move among different receiving zones. This is a popular example for continuous switched systems =-=[14]-=-, yet many properties of the system can be specified in temporal logic. For example, we would like to check that whenever a network available signal appears, it stays valid as long as the cell phone d... |

122 | Decidability of second order theories and automata on infinite trees - Rabin - 1969 |

111 |
On the complexity of ω-automata
- Safra
- 1988
(Show Context)
Citation Context ...es ψ. We reduced the switching-rule synthesis problem to the nonemptiness problem for . The number of states of the DPW Aψ is doubly-exponential in |ψ|, and its A T1,T2 ∀ψ index is exponential in |ψ| =-=[20, 23]-=-. Therefore, the number of states of the DPT A T1,T2 ∀ψ is linear in |T1| and |T2| and doubly-exponential in |ψ|, and its index is exponential in |ψ|. Since the nonemptiness problem for DPT can be sol... |

92 |
Fundamentals of Power Electronics
- Erickson, Maksimovic
- 2001
(Show Context)
Citation Context ... Examples to switched systems include software systems (c.f., internet communication protocols [7, 22]), mechanical systems (engines with gear transmission [8]), electrical circuits (power converters =-=[6]-=-), biological systems (gene regulating networks [3]), and embedded systems combining the above [2]. There has been extensive research in the control engineering community on analysis of continuous swi... |

87 | O-minimal hybrid systems
- Lafferriere, Pappas, et al.
- 1998
(Show Context)
Citation Context ...e crucial in the infinite-state setting. For example, reachability in o-minimal hybrid systems is decidable only when each discrete control state has a single initial value for the continues elements =-=[11]-=-. Obviously, however, resuming the interaction from a fixed state is a much weaker composition mechanism. Remark 2. In [15], Mayer and Stockmeyer studied the complexity of membership and inequality fo... |

86 | Lower bounds for natural proof systems - KOZEN - 1977 |

79 |
Distributed reactive systems are hard to synthesize
- Pnueli, Rosner
- 1990
(Show Context)
Citation Context ...de whether there are transducers T1, T2, . . . , Tn such that Ti satisfies ϕi for all 1 ≤ i ≤ n, and T1 ⊕ T2 ⊕ · · · ⊕ Tn satisfies ψ. On the negative side, we show that, as with standard concurrency =-=[18]-=- compositional-realizability is undecidable. Sometimes, the details of the switching mechanism are known and may be controlled. On the positive side, we study the problem of synthesizing a switching r... |

56 |
transition from global to modular temporal reasoning about programs
- In
(Show Context)
Citation Context ...ducers, and let ϕ1, ϕ2, and ϕ3 be LTL formulas. Below are two typical assume-guarantee rules, for a composition operator ⊕ ∈ {�� , �� } (as with the known composition semantics, many more rules exist =-=[16]-=-). 〈ϕ1〉T1〈ϕ2〉 〈ϕ2〉T2〈ϕ3〉 〈ϕ1〉T1 ⊕ T2〈ϕ3〉 〈true〉T1〈ϕ1〉 〈true〉T2〈ϕ2〉 〈true〉T1 ⊕ T2〈ϕ1 ∧ ϕ2〉 Consider for example the left rule. To see that this rule is sound, note that, by definition, for every T ′ we... |

26 | Hybrid modeling of TCP congestion control
- Hespanha, Bohacek, et al.
- 2001
(Show Context)
Citation Context ...defined above, may also serve as an abstraction of other, not necessarily finite-state, switched systems. Examples to switched systems include software systems (c.f., internet communication protocols =-=[7, 22]-=-), mechanical systems (engines with gear transmission [8]), electrical circuits (power converters [6]), biological systems (gene regulating networks [3]), and embedded systems combining the above [2].... |

25 |
Stability analysis of switched systems using variational principles: An introduction
- Margaliot
- 2006
(Show Context)
Citation Context ...ach moment in time one of the components determines the behavior of the system, while the other components are ignored. Such a “switching semantics” has been well-studied in the engineering community =-=[12, 13]-=-. In this paper, we study it for finite-state systems. Given finite-state transducers T1, T2, . . . , Tn, all interacting with the same environment, we define the switched system T1 ⊕ T2 ⊕ · · · ⊕ Tn ... |

23 | The complexity of word problems - this time with interleaving
- Mayer, Stockmeyer
- 1994
(Show Context)
Citation Context ...is possible to model the dormant and active compositions using known synchronous and asynchronous composition operators. Such a modeling, however, is less clean, and hides the switching mechanism. In =-=[15]-=-, Mayer and Stockmeyer studied regular expressions extended with a shuffle operator on words, which interleaves its operands. As we show later, the shuffle operator corresponds to a dormant compositio... |

17 |
Special issue on hybrid systems: Theory and applications-a brief introduction to the theory and application of hybrid systems
- Antsaklis
(Show Context)
Citation Context ... 22]), mechanical systems (engines with gear transmission [8]), electrical circuits (power converters [6]), biological systems (gene regulating networks [3]), and embedded systems combining the above =-=[2]-=-. There has been extensive research in the control engineering community on analysis of continuous switched systems whose evolution is described by means of differential equations [12, 13]. The study ... |

17 | temporal logics - Automata - 1985 |

17 | Exponential Determinization for ω-Automata with Strong-Fairness Acceptance Condition - Safra - 1992 |

17 | Mathematical puzzles: a connoisseur's collection - Winkler - 2004 |

14 | Automata based interfaces for control and scheduling
- Weiss, Alur
- 2007
(Show Context)
Citation Context ... of the paper). When we synthesize a switching rule, we are given the transducers T1 and T2, and the transducer we are after only has to generate an infinite sequence over {1, 2}. The 7 A recent work =-=[24]-=- advocates the use of ω-regular languages over the alphabet of subcomponents identifiers for describing switching constraints even for continuous switched systems.setting is then similar to the contr... |

8 |
Analysis and design of AIMD congestion control algorithms in communication networks
- Shorten, Leith, et al.
- 2005
(Show Context)
Citation Context ...defined above, may also serve as an abstraction of other, not necessarily finite-state, switched systems. Examples to switched systems include software systems (c.f., internet communication protocols =-=[7, 22]-=-), mechanical systems (engines with gear transmission [8]), electrical circuits (power converters [6]), biological systems (gene regulating networks [3]), and embedded systems combining the above [2].... |

6 | Abstract patterns for compositional reasoning
- Amla, Emerson, et al.
- 2003
(Show Context)
Citation Context ...e concurrent system has fewer behaviors than its components. The fact that the composed system has fewer behavior than its components has played a central role in compositional reasoning. As shown in =-=[1]-=-, both synchronous and asynchronous parallel compositions can be seen as intersection of the enhanced language of its components. Further classes of parallel compositions have been studied in [1]. The... |

2 |
Decision problems for omega-automata
- Landweber
- 1969
(Show Context)
Citation Context ...unting. As detailed in the full version, we do this by first proving that if L ⊆ Σ ω is preserved under merge and is regular, then L is accepted by a deterministic co-Büchi automaton. Essentially, in =-=[10]-=-, Landweber proves that a deterministic Rabin automaton has an equivalent deterministic Büchi automaton iff its accepting strongly connected components are upward closed (that is, if S is accepting, s... |

1 |
Modeling of hybrid systems. Encyclopedia of life support systems
- Johansson, Lygeros, et al.
- 2004
(Show Context)
Citation Context ... necessarily finite-state, switched systems. Examples to switched systems include software systems (c.f., internet communication protocols [7, 22]), mechanical systems (engines with gear transmission =-=[8]-=-), electrical circuits (power converters [6]), biological systems (gene regulating networks [3]), and embedded systems combining the above [2]. There has been extensive research in the control enginee... |